Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets

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https://doi.org/10.1007/s10898-020-00927-7

Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets

N. T. T. Huong¹·J.-C. Yao²·N. D. Yen³

Received: 8 July 2019 / Accepted: 13 July 2020

Abstract

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with aboundedconstraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems withunboundedconstraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

Keywords Linear fractional vector optimization·Efficient solution·Gain-to-loss ratio· Geoffrion’s properly efficient solution·Unbounded constraint set·Direction of recession· Regularity condition

Mathematics Subject Classification 90C29·90C32·49K30·49N60

1 Introduction

Linear fractional vector optimization problems(LFVOPs) have many applications in management science. Due to their remarkable properties and theoretical importance, these problems

The authors were supported by National Foundation for Science and Technology Development (Vietnam) under Grant No. 101.01-2018.306, Le Quy Don Technical University (Vietnam), Grant MOST

105-2115-M-039-002-MY3 (Taiwan), and the Vietnam Institute for Advanced Study in Mathematics (VIASM, Vietnam).

B

^{N. D. Yen}

[email protected]

Extended author information available on the last page of the article

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have been studied intensively. Steuer [30, p. 337] has observed that linear fractional (ratio) criteria are frequently encountered in finance: min{debt-to-equity ratio}, max{return to investment}, max{output per employee}, min{actual cost to standard cost} (in Corporate Planning); min{risk-assets to capital}, max{actual capital to required capital}, min{foreign loans to total loans}, min{residental mortgages to total mortgages} (in Bank Balance Sheet Management). Fractional objectives also occur in other areas of management: transportation management, education management, and medicine management (see, e.g., [30]).

LFVOPs constitute a special class of multiobjective fractional optimization problems.

Note that, among the 520 articles cited by Stancu-Minasian in his ninth bibliography of fractional programming [29], 38 papers are devoted to multiobjective fractional programming.

Connections of LFVOPs withmonotone affine vector variational inequalitieswere firstly recognized by Yen and Phuong [32].

Topological properties of the solution sets of LFVOPs and monotone affine vector variational inequalities have been studied by Choo and Atkins [8,9], Benoist [1,2], Yen and Phuong [32], Hoa et al. [17–19], Huong et al. [20,21], and other authors. In particular, the authors of [17] showed thatfor any positive integer m,there exists a LFVOP with m objective criteria which has exactly m connected components in the Pareto efficient solution set and in the weak efficient solution set.Recently, based on some theorems from real algebraic geometry [4], we have been able to prove in [21] thatthe number of the connected components in the Pareto efficient solution set (resp., in the weak efficient solution set) of any LFVOP is finite.

From a fundamental theorem of Robinson [26, Theorem 2] on stability of monotone affine variational inequalities, Yen and Yao [34, Section 5] have derived several results on solution stability and topological properties of the solution sets of LFVOPs. Numerical methods for solving LFVOPs can be found in Steuer [30] and Malivert [25]. The interested readers are referred to the survey paper of Yen [31] for more information about linear fractional and convex quadratic vector optimization problems.

Very recently, Yen and Yang [33] have initiated a study on infinite-dimensional LFVOPs and infinite-dimensional convex quadratic vector optimization problems via affine variational inequalities on normed spaces.

Open questions concerning qualitative properties of finite-dimensional LFVOPs remain.

For instance,one still cannot prove or disprove the conjecture saying that the number of the connected components in the Pareto efficient solution set (resp., in the weak efficient solution set) of a LFVOP with m objective criteria does not exceed m.

Considering a general vector optimization problem, Geoffrion [13] noticed that certain efficient solutions show unfavorable properties concerningthe ratio of the rate of gain in one cost over the corresponding rate of loss in another cost. To clarify this kind of pathological behavior of the efficient solutions, he introduced [13, pp. 618–619] the notionproperly efficient solution. Later, several extensions of Geoffrion’s concept of proper efficiency were given by Borwein [5], Benson [3], Henig [16], and other scholars. Discussions on the rela- tionships among different types of properly efficient solutions can be found in the papers of Benson [3] and of Guerraggio et al. [15].

Probably the paper by Choo [7] is the first work addressing Geoffrion’s concept of proper efficiency for LFVOPs. We are indebted to one referee of an earlier version of the present paper for making us familiar with the significant work [7], from which we have learned that any efficient solution of a linear fractional vector optimization withboundedconstraint set is properly efficient in the sense of Geoffrion. The proof of this remarkable theorem is enough complicated. To be more precise, Choo has used several technical lemmas and original arguments based on the necessary and sufficient conditions for a feasible point of a LFVOP to be an efficient solution, which will be recalled in Sect.2below.

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Chew and Choo [6, Example 3.6] showed that there exits a LFVOP with an unbounded constraint set where none of the efficient solutions is properly efficient in the sense of Geof- frion. Note that the efficient solution set of the problem coincides with the constraint set.

Apart from the just-cited example, up to now one still does not have any further information about the difference between the efficient solution set and the Geoffrion properly efficient solution set of a LFVOP with an unbounded constraint set.

By constructing suitable examples, we will show that there exist LFVOPs whose efficient solution sets do not coincide with the unbounded constraint sets, which have improperly efficient solutions (see Examples2.6and4.7below). Our main results give sufficient conditions for an efficient solution of a LFVOP with an unbounded constraint set to belong to its properly efficient solution set. These conditions require certain regularity conditions, where the recession cone of the constraint set appears in a natural way. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. [17].

The variational inequality characterization of the efficient solutions of LFVOPs, which can be interpreted [32] as the approach to LFVOPs via the concept ofvector variational inequality(VVI for brevity), is the starting point of this research. In Proposition4.2below, by using this characterization, we are able to prove that if the denominators of the objective functions are all the same, then every efficient solution is a Geoffrion’s properly efficient solution. The relations between the Pareto solutions of a VVI and the proper efficiency have been explored by Crespi [10]. Since the Klinger properly efficient solution set of a LFVOP coincides with its efficient solution set, from Theorems 5–7 of [10] one recovers the known result [32, Remark 2] saying that the efficient solution set of LFVOP coincides with the Pareto solution set of the corresponding affine VVI. Furthermore, from [13, Theorem 1]

and [10, Theorem 1] it follows that the Hurwicz properly efficient solution set of a LFVOP is a subset of its Geoffrion properly efficient solution set. Applied to a LFVOP, Theorem 11 of [10] says that if all the objective functions of the problem are convex on the whole space (this happens if and only if all the functions are affine), then the Benson properly efficient solution set of the given LFVOP coincides with its Hurwicz properly efficient solution set.

As the former solution set is equal to the Geoffrion properly efficient solution set, this result follows from [13, Theorem 2]. Thus, the results of the present paper cannot be derived from the results of Crespi [10].

The contents of the rest of the paper are as follows. Section2recalls some definitions, auxiliary results, and presents a pathological bicriteria LFVOP which has improperly efficient solutions. Section3establishes the main results. They are analyzed in Sect.4where, among other things, a three-criteria LFVOP having improperly efficient solutions is given. The last section is devoted to some remarks and open questions.

2 Preliminaries

Forx¹,x²,x inRⁿ, the scalar product of the first two vectors and the Euclidean norm of the third one are denoted, respectively, byx¹,x²andx. Vectors inRⁿare interpreted as columns of real numbers in matrix calculations but, for simplicity, sometimes they will be described by rows of real numbers. The transpose of a matrixAis denoted by A^T. ByNwe denote the set of the positive integers.

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Considerlinear fractional functions f_i:Rⁿ →R,i =1, . . . ,m, of the form f_i(x)= a_i^Tx+αi

b_i^Tx+βi

,

whereai ∈ Rⁿ,bi ∈ Rⁿ, αi ∈ R,andβi ∈ R. LetK be apolyhedral convex set, i.e., there exist p ∈ N, a matrixC = (ci j) ∈ R^p×n,and a vectord = (di) ∈ R^p such that K =

x ∈Rⁿ : C x ≤d

. Our standing condition is thatb_i^Tx+βi >0 for alli ∈I and x ∈K, whereI := {1, . . . ,m}. Put f(x)=(f₁(x), . . . ,f_m(x))and let

=

x∈Rⁿ : b_i^Tx+βi>0, ∀i ∈I .

Clearly,is open and convex,K ⊂, and f is continuously differentiable on. Thelinear fractional vector optimization problem(LFVOP) given by f andK is formally written as

(VP) Minimize f(x) subject to x ∈K.

Definition 2.1 A point x ∈ K is said to be anefficient solution(or aPareto solution) of (VP) if

f(K)− f(x)

∩

−R^m₊\{0}

= ∅, whereR^m₊ denotes the nonnegative orthant inR^m. One callsx ∈ K aweakly efficient solution(or aweak Pareto solution) of(VP)if f(K)− f(x)

∩

−intR^m₊

= ∅, where intR^m₊abbreviates the topological interior ofR^m₊. The efficient solution set (resp., the weakly efficient solution set) of (VP) are denoted, respectively, byEandE^w. According to [8,25] (see also [23, Theorem 8.1]),for any x∈K, one has x ∈ E (resp., x ∈ E^w)if and only if there exists a multiplierξ =(ξ1, . . . , ξm) ∈ intR^m₊(resp.,ξ=(ξ1, . . . , ξm)∈R^m₊\{0}) such that

^m

i=1

ξi

b_i^Tx+βi

a_i−

a_i^Tx+αi)b_i ,y−x

≥0, ∀y∈K. (2.1)

Ifbi =0 andβi =1 for alli ∈I, then (VP) coincides with the classicalmultiobjective linear optimization problem(see Luc [24] and the references therein). By the above optimality conditions, for anyx∈K, one hasx ∈E(resp.,x ∈E^w) if and only if there exists a multiplier ξ=(ξ1, . . . , ξm)∈intR^m₊(resp., a multiplierξ =(ξ1, . . . , ξm)∈R^m₊\{0}) such that

^m

i=1

ξiai,y−x

≥0, ∀y∈K. (2.2)

During the last four decades, LFVOPs have attracted a lot of attention from researchers;

see [7–9,17–22,25,30–34], Chapter 8 of [23], and the references therein. By the results of Choo and Atkins [9], Benoist [1], Yen and Phuong [32], one knows that ifK bounded, then both solution setsEandE^wof (VP) are connected. Later, Hoa et al. [17] showed that ifKis noncompact, then both solution setsEandE^wof (VP) wherem=ncan havemconnected components. Stability properties of (VP) with f andKbeing subject to perturbations can be found in [34]. Recently, by using some tools from real algebraic geometry, Huong et al. [21, Theorem 4.5] have proved that bothEandE^ware semi-algebraic sets inRⁿ; hence they have finitely many connected components. However, as far as we know, the proper efficiency of the solutions of (VP) has been studied just in the papers by Choo [7], Chew and Choo [6].

There are several notions of proper efficiency in vector optimization. The most fundamental ones have been suggested by Geoffrion [13], Borwein [5], Benson [3], and Henig [16]. It is worthy to stress that the notion of properly efficient solution proposed by Geoffrion [13] in 1968 has a clear practical meaning in the form of again-to-loss ratio[13, p. 624] and it is the

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starting point for all subsequent studies on proper efficiency. Applied to (VP), Geoffrion’s definition of properly efficient solution can be formulated as follows.

Definition 2.2 (See [13, p. 618]) One says thatx¯ ∈ E is aGeoffrion’s properly efficient solutionof(VP)if there exists a scalar M>0 such that, for eachi ∈ I, wheneverx ∈ K andfi(x) < fi(x¯)one can find an indexj∈Isuch that fj(x) > fj(x)¯ andAi,j(x¯,x)≤M withA_i,j(x¯,x):= f_i(x¯)− f_i(x)

f_j(x)− f_j(x¯).

Geoffrion’s efficient solution set of (VP) is denoted byE^Ge. Geoffrion [13, p. 619] noticed the following:For anyx¯ ∈ E,x¯ ∈/ E^Geif and only if for every scalar M >0there exist x ∈ K and i ∈ I with fi(x) < fi(x¯)such that, for all j ∈ I satisfying fj(x) > fj(x¯), one has A_i,j(x¯,x) >M. This observation was made in [13] for general vector optimization problems which may not belong to the class of LFVOPs.

Remark 2.3 The quantity A_i,j(x¯,x)given in Definition2.2is a gain-to-loss ratio. Namely, it is the ratio ofthe gain fi(x¯)− fi(x) >0 of the planxover the planx¯in theith criterion bythe loss fj(x)− fj(x¯) >0 ofx overx¯in the jth criterion.

Remark 2.4 If (VP) is alinear vector optimization problem(that is,bi =0 andβi =1 for alli ∈I), thenE^Ge =E. This well-known result can be proved easily by using Theorem 1 from [13] and the optimality condition (2.2). Indeed, ifx∈E, then there exists a multiplier ξ=(ξ1, . . . , ξm)∈intR^m₊such that (2.2) holds. So,

^m

i=1

ξia_i,y

≥^m

i=1

ξia_i,x

, ∀y∈K.

Since f_i(x)=a_i^Tx+αi for alli∈I, this implies that m

i=1

ξif_i(y)≥ m i=1

ξif_i(x), ∀y∈K.

Therefore, by [13, Theorem 1] one has x ∈ E^Ge. As the inclusion E^Ge ⊂ E follows from Definition2.2, we have thus proved thatE^Ge=E.

As shown in the following example and some examples in Sect.4, the equalityE^Ge =E still hods for many LFVOPs, which do not belong to the class of linear vector optimization problems.

Example 2.5 Consider problem (VP) with m = 2, n = 1, K =

x ∈ R : x ≥ ¹₂ , f1(x) = x⁻¹ and f2(x) = x for allx ∈ K. It is easy to verify thatE = ₁

2,+∞

.We will show thatE^Ge= E. Take an arbitrary pointx¯ ∈Eand putM =max

x¯⁻²,x¯²

. Ifx belongs to the set{x∈K : x>x¯}, then fori =1 and j=2 one has

A_i,j(x,¯ x)= x¯⁻¹−x⁻¹ x− ¯x = 1

xx¯ ≤ 1

¯ x² ≤M.

Similarly, ifx belongs to the set{x∈K : x <x¯}, then fori=2 and j =1 one has Ai,j(x¯,x)= x¯−x

x⁻¹− ¯x⁻¹ =xx¯≤ ¯x²≤M.

This proves thatx¯∈E^Ge; henceE^Ge=E.

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In addition to the example of Chew and Choo [6] cited in Sect.1, the next example signifies the existence of pathological LFVOPs which have Geoffrion’s improperly efficient solutions.

Example 2.6 Consider the problem (VP) with K =

x=(x₁,x₂)∈R² : x₁≥0, x₂≥0 , f1(x)= −x2, f2(x)= x2

x₁+x₂+1.

Using the necessary and sufficient condition for the efficiency in (2.1), one can show that E = {(x1,0) : x1 ≥ 0}.Select any vector x¯ = (α,0), α ≥ 0,from E. We will have

¯

x ∈/ E^Ge, iffor every scalar M >0there exist x∈ K andi ∈I with f_i(x) < f_i(x¯)such that, for all j∈I satisfying f_j(x) > f_j(x¯),

A_i,j(x¯,x)= fi(x¯)− fi(x) fj(x)− fj(x¯) >M.

Given any M > 0, we choose i = 1 and select a pointx = (x1,x2) ∈ K with x1 ≥ 0,x2>0,x1+x2+1> M. Then, fi(x) < fi(x)¯ and the unique index j ∈ Isatisfying

f_j(x) > f_j(x¯)is j=2. We have

A_1,2(x¯,x)= 0−(−x₂) x₂

x1+x2+1− 0 α+1

=x₁+x₂+1>M.

Thus, all the efficient points are improperly efficient in the sense of Geoffrion. In other words, E^Ge= ∅, whileEis unbounded.

In the sequel, to establish verifiable sufficient conditions for a pointx¯ ∈ Eto belong to E^Ge, we will need the forthcoming lemma.

Lemma 2.7 (See, e.g., [25] and [23, Lemma 8.1])Letϕ(x)= a^Tx+α

b^Tx+β be a linear fractional function defined by a,b∈Rⁿandα, β ∈R. Suppose that b^Tx+β=0for every x ∈K0, where K₀⊂Rⁿis an arbitrary convex set. Then, one has

ϕ(y)−ϕ(x)=b^Tx+β

b^Ty+β∇ϕ(x),y−x, (2.3)

for any x,y∈K₀, where∇ϕ(x)denotes the Fréchet derivative ofϕat x.

The connectedness ofK0 and the conditionb^Tx +β =0 for everyx ∈ K0 imply that eitherb^Tx+β >0 for allx ∈K₀, orb^Tx+β <0 for allx∈K₀. Hence, for anyx,y∈K₀, one has b^Tx+β

b^Ty+β >0. Given vectorsx,y ∈K₀withx =y,we consider two points from the line segment[x,y]:

zt =x+t(y−x), z_t=x+t(y−x) (t∈ [0,1],t∈ [0,t)).

By (2.3) we can assert that

(i) If∇ϕ(x),y−x>0, thenϕ(zt) < ϕ(zt)for everyt∈ [0,t);

(ii) If∇ϕ(x),y−x<0, thenϕ(zt) > ϕ(zt)for everyt∈ [0,t);

(iii) If∇ϕ(x),y−x =0, thenϕ(zt)=ϕ(zt)for everyt∈ [0,t).

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This shows thatϕis monotonic on every line segment or ray contained inK₀.

Remark 2.8 The monotonicity of a linear fractional function on every line segment or ray contained in its effective domain has some connection with the concept of increasing convex- along-rays functions [28, Definition 2.1], which is defined for functions fromRⁿ₊toR. This concept is useful for global optimality conditions for a class of nonconvex optimization problems [28, Section 4]. Note that a linear fractional function defined onRⁿ₊ need not to be an increasing convex-along-rays function. Crepsi et al. [11] showed that star-shaped set and increasingly along rays functions have interesting applications in Minty variational inequalities. In a subsequent paper [12, Definition 4.1], they showed how the concept of increasingly along rays function [11, Definition 3] can be extended for vector functions.

To deal with LFVOPs having unbounded constraint sets, we will use the notion of recession cone. Recall [27, p. 61] that a nonzero vectorv∈Rⁿis said to be adirection of recessionof a nonempty convex setD ⊂Rⁿ ifx+tv∈ Dfor everyt ≥0 and everyx ∈ D.The set composed by 0∈Rⁿand all the directionsv∈Rⁿ\{0}satisfying the last condition, is called therecession coneofDand denoted by 0⁺D.IfDis closed and convex, then

0⁺D= {v∈Rⁿ : ∃x ∈ s.t. x+tv∈D for all t>0}. (2.4) The recession cone of a polyhedral convex set can be easily computed by using the next lemma, which can be proved by a direct verification.

Lemma 2.9 If K =

x ∈Rⁿ : C x ≤d

with C ∈R^p×nand d ∈R^p, and K is nonempty, then0⁺K =

v∈Rⁿ : Cv≤0 .

We will need the following lemma, whose elementary proof is given for the sake of completeness.

Lemma 2.10 Let C be a closed and convex set inRⁿ,x¯∈C, and let{x^k}be a sequence in C\{ ¯x}with lim

k→∞x^k = +∞. If lim

k→∞

x^k− ¯x

x^k− ¯x =v, thenv∈0⁺C.

Proof By (2.4), to obtain the desired conclusion we need to show thatx¯+tv ∈C for an arbitrarily givent > 0. Choosek₁ such thatx^k− ¯x > t for allk ≥ k₁. Then, by the convexity ofC, vector

¯

x+t x^k− ¯x

x^k− ¯x =x^k− ¯x −t

x^k− ¯x x¯+ t x^k− ¯xx^k belongs toC for everyk ≥k₁. Passing the expressionx¯+t x^k− ¯x

x^k− ¯x to limit ask → ∞

and using the closedness ofC, one getsx¯+tv∈C.

3 Sufficient conditions for Geoffrion’s proper efficiency First, we study Geoffrion’s proper efficiency for bicriteria LFVOPs.

Theorem 3.1 Suppose that m=2andx¯∈E. Ifthe first regularity condition

t her e exi st no(i,j)∈I², j=i, andv∈(0⁺K)\{0}wi t h

∇fi(x), v =¯ 0 and

∇fj(x¯), v

=0 (3.1)

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andthe second regularity condition

t her e exi st no(i,j)∈I², j =i, andv∈(0⁺K)\{0}such t hat b_i^Tv=0, ∇f_i(x¯), v ≤0,

∇f_j(x¯), v

>0 (3.2)

are satisfied, thenx¯∈E^Ge.

Proof Suppose that (3.1) and (3.2) hold at a pointx¯∈E, butx¯∈/E^Ge. Then, for everyp∈N, there existx^p∈K andi(p)∈Iwith fi(p)(x^p) < f_i(p)(x)¯ such that, for all j∈Isatisfying fj(x^p) > fj(x¯), one hasAi(p),j(x¯,x^p) > p(see the observation after Definition2.2). Since the sequence{i(p)}has values in the finite setI, by choosing a subsequence, we may assume thati(p)=ifor allp, wherei∈Iis a fixed index. For eachp, asx¯∈Eandf_i(x^p) < f_i(x¯), there must exist an index j(p)∈I\{i}satisfying f_j(p)(x^p) > f_j(_p)(x¯). Since the sequence {j(p)}has values in the finite setI\{i}, by considering a subsequence, we may assume that

j(p)= jfor all p, where j∈I\{i}is a fixed index. Thus, Ai,j(x¯,x^p)= fi(x¯)− fi(x^p)

fj(x^p)− fj(x¯) > p (∀p∈N). (3.3) If{x^p}is bounded, then we can obtain a contradiction by arguing as in the proof of the main theorem of Choo [7, p. 220].

If{x^p}is unbounded then, by selecting a subsequence (if necessary), we may assume that

plim→∞x^p = +∞. Letv^p = x^p− ¯x

x^p− ¯x. Without loss of generality, we can suppose that

plim→∞v^p=v, wherevis a unit vector. SinceK is closed and convex, applying Lemma2.10 givesv∈(0⁺K)\{0}. By Lemma2.7,

f_i(x¯)− f_i(x^p)= − b_i^Tx¯+βi

b_i^Tx^p+βi

∇f_i(x¯),x^p− ¯x

= −x^p− ¯xb_i^Tx¯+βi

b_i^Tx^p+βi

∇f_i(x¯), v^p .

(3.4)

Similarly,

fj(x^p)− fj(x)¯ = b^T_jx¯+βj

b^T_jx^p+βj

∇fj(x¯),x^p− ¯x

= x^p− ¯x b^T_jx¯+βj

b^T_jx^p+βj

∇fj(x¯), v^p .

(3.5)

Hence,

Ai,j(x¯,x^p)= − fi(x^p)− fi(x¯) fj(x^p)− fj(x¯)

= −b_i^Tx¯+βi

b_i^Tx^p+βi

.b^T_jx^p+βj

b^T_jx¯+βj

.∇ f_i(x¯), v^p

∇f_j(x), v¯ ^p.

Sinceb_k^Tx¯+βk >0 for everyk ∈I, one sees thatAi,j(x,¯ x^p)tends to+∞if and only if the quantity

A¯_i,j(x¯,x^p):= −b^T_jx^p+βj

b^T_i x^p+βi

.∇ fi(x¯), v^p

∇f_j(x¯), v^p

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tends to+∞asp→ ∞. Note that b^T_jx^p+βj

x^p− ¯x =b^T_j(x^p− ¯x)

x^p− ¯x + βj

x^p− ¯x+ b^T_jx¯

x^p− ¯x (3.6)

and

b_i^Tx^p+βi

x^p− ¯x =b_i^T(x^p− ¯x)

x^p− ¯x + βi

x^p− ¯x+ b^T_i x¯

x^p− ¯x. (3.7)

Therefore,

A¯_i,_j(x¯,x^p)= −

b^T_j(x^p− ¯x)

x^p− ¯x + βj

x^p− ¯x+ b^T_jx¯ x^p− ¯x b_i^T(x^p− ¯x)

x^p− ¯x + βi

x^p− ¯x+ b_i^Tx¯ x^p− ¯x

.∇ f_i(x¯), v^p

∇f_j(x¯), v^p. Since lim

p→∞x^p = +∞, one has lim

p→∞x^p− ¯x = +∞. Thus, if(b^T_i v)

∇fj(x¯), v

=0, then

plim→∞A¯_i_,j(x¯,x^p)= −(b^T_jv)∇fi(x¯), v (b^T_i v)

∇fj(x¯), v. (3.8) This contradicts the condition lim

p→∞A¯_i,j(x¯,x^p)= +∞. So, the denominator of the fraction in (3.8) must be 0, i.e.,

(b_i^Tv)

∇fj(x), v¯

=0, (3.9)

wherev∈(0⁺K)\{0}.

Sinceb^T_kx+βk >0 for allk ∈ Iandx ∈ K, one has b^T_jx^p+βj

x^p− ¯x >0 for all p ∈N.

Hence, passing (3.6) to limit as p→ ∞, one obtainsb^T_jv≥0. As f_i(x¯)− f_i(x^p) >0 for everyp∈N, from (3.4) it follows that∇f_i(x¯), v^p<0 for all p∈N. Hence,

b^T_jv≥0 and ∇fi(x), v ≤¯ 0.

Analogously, from (3.7) and (3.5) one can deduce that b^T_i v≥0 and

∇fj(x), v¯

≥0.

Since thedenominatorof the fraction in (3.8) equals to 0 by (3.9), one of the following situations must occur:

(d1) b_i^Tv=0 and

∇fj(x¯), v

=0;

(d2) b_i^Tv >0 and

∇fj(x¯), v

=0;

(d3) b_i^Tv=0 and

∇fj(x¯), v

>0.

As thenumeratorof (3.8) can be either zero or a negative number, there are four possibilities:

(n₁) b^T_jv=0 and∇f_i(x¯), v =0;

(n₂) b^T_jv=0 and∇f_i(x), v¯ <0;

(n3) b^T_jv >0 and∇fi(x), v =¯ 0;

(n₄) b^T_jv >0 and∇f_i(x¯), v<0.

If (d1) and (n1) occur, then one has

∇fj(x¯), v

=0 and∇fi(x), v =¯ 0. This contradicts condition (3.1), becausev∈(0⁺K)\{0}.

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If (d₁) and (n₂) take place, then one has

∇f_j(x¯), v

= 0 and ∇f_i(x¯), v < 0. Since v ∈ (0⁺K)\{0},x¯+tv ∈ K for everyt ≥ 0. Therefore, thanks to Lemma2.7, we have f_j(x¯+tv)= f_j(x¯)and f_i(x¯+tv) < f_i(x¯)for everyt>0. Asm=2, this contradicts the assumptionx¯∈E.

The case where (d1) and (n3) (resp., (d1) and (n4)) occur can be treated similarly as that where (d₁) and (n₁) (resp., (d₁) and (n₂)) occur.

It is clear that the situations (d₂)–(n₁), (d₂)–(n₂), (d₂)–(n₃), and (d₂)–(n₄) also lead to a conflict either with the first regularity condition or with the inclusionx¯∈E.

Finally, observe that each one of the situations (d3)–(n1), (d3)–(n2), (d3)–(n3), and (d3)–

(n4) leads to the system

b^T_i v=0, ∇fi(x¯), v ≤0,

∇fj(x), v¯

>0,

wherev∈(0⁺K)\{0}. This obviously contradicts the regularity condition in (3.2).

We have thus established the inclusionx¯∈E^Geand completed the proof.

Consider the problem in Example2.6. For every efficient solutionx¯=(x¯₁,0), we have

∇f1(x¯)=(0,−1)^T, ∇f2(x¯)= 0

¯1 x1+1

.

Hence, (3.1) is violated if one chosesi=1, j=2 andv=(1,0)∈(0⁺K)\{0}. Moreover, condition (3.2) is also violated fori =1, j =2 andv=(v1, v2)withv1 ≥0 andv2>0.

Here, the violation of the regularity conditions of Theorem3.1is a reason forx¯∈/ E^Ge. Now, we study Geoffrion’s proper efficiency for LFVOPs having more than two objective functions.

Theorem 3.2 In the case m≥3, suppose thatx¯∈E. If(3.1),(3.2), andthe third regularity condition

t her e exi st no tr i plet(i,j,k)∈I³, wher e i,j,k ar e pairwi se di sti nct, andv∈(0⁺K)\{0}wi t h∇fi(x¯), v<0,

∇fj(x¯), v

=0,∇fk(x), v¯ >0 (3.10) are satisfied, thenx¯∈E^Ge.

Proof Suppose to the contrary that x¯ ∈ E and (3.1), (3.10), and (3.2) are satisfied, but

¯

x ∈/ E^Ge. Then, as it has been shown in the proof of Theorem3.1, there exist a sequence {x^p} ⊂K\{ ¯x}and a pair(i,j)∈I²,i= j, such that (3.3) holds.

If{x^p}is bounded, then the arguments used for proving the main theorem of Choo [7, p. 220] lead us to a contradiction.

If{x^p}is unbounded then, we may assume that lim

p→∞x^p = +∞. Arguing as in the proof of Theorem3.1, we can find a vectorv∈0⁺K withv =1, such that thedenominatorof the fraction in (3.8) equals to 0. Hence, one of the three situations (d1)–(d3) must happen. As thenumeratorof (3.8) can be either zero or a negative number, one of the four possibilities (n1)–(n4) occurs.

The situations (d₁)–(n₁), (d₁)–(n₃), (d₂)–(n₁), (d₂)–(n₃) are prohibited by condition (3.1).

The situations (d₃)–(n₁), (d₃)–(n₂), (d₃)–(n₃), (d₃)–(n₄) are excluded by condition (3.2).

Indeed, in each one of these situations, one hasb^T_i v=0,∇fi(x¯), v≤0, and

∇fj(x¯), v

>0.

Consider the situation (d1)–(n2). Asv ∈ (0⁺K)\{0}, one has x¯+tv ∈ K for every t ≥ 0. Since∇fi(x¯), v < 0 and

∇fj(x¯), v

= 0, by Lemma2.7we can assert that f_i(x¯+tv) < f_i(x)¯ and f_j(x¯+tv)= f_j(x¯)for everyt>0. Hence, noting that 0⁺K\{0},

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from (3.10) we have ∇f_k(x¯), v ≤ 0 for every k ∈ I\{i,j}. Therefore, according to Lemma2.7, f_k(x¯+tv)≤ f_k(x¯)for anyk ∈I\{i,j}andt>0. Picking anyt∈(0,+∞), we observe that the inequalities f_k(x¯+tv) ≤ f_k(x¯) for allk ∈ I\{i}together with the strict inequality fi(x¯+tv) < fi(x)¯ contradict the inclusionx¯∈E. If one of the situations (d2)–(n2), (d2)–(n4), and (d1)–(n4) occurs, we can obtain a contradiction by arguing similarly.

The proof is complete.

4 Further analysis

4.1 Choo and Atkins’ example

The sufficient conditions for Geoffrion’s proper efficiency given in Sect.3are verifiable. To justify this assertion, we consider the well-known LFVOP having disconnected solution sets, which was given by Choo and Atkins [9].

Example 4.1 (See [9, Example 2]) Consider problem (VP) with K =

x=(x1,x2)∈R² : x1≥2, 0≤x2≤4 , f₁(x)= −x₁

x₁+x₂−1, f₂(x)= −x₁ x₁−x₂+3.

Using the criteria forx ∈Eandx∈E^wrecalled in (2.1), it is easy to show that E=E^w=

(x₁,0) : x₁≥2} ∪ {(x₁,4) : x₁≥2 .

Here we haveb1 =(1,1),b2 =(1,−1),and 0⁺K = {v=(v1,0) : v1 ≥0}.For every

¯

x ∈ {(x¯₁,0) : ¯x₁≥2},

∇f₁(x¯)=

⎛

⎜⎝ 1 (x¯₁−1)²

¯ x₁ (x¯₁−1)²

⎞

⎟⎠, ∇f₂(x¯)=

⎛

⎜⎝

−3 (x¯₁+3)²

− ¯x₁ (x¯₁+3)²

⎞

⎟⎠.

Hence, for any vectorv=(v1, v2), ∇f1(x¯), v =0

∇f₂(x¯), v =0 ⇐⇒

v1+ ¯x1v2 =0

−3v1+ ¯x₁v2 =0 ⇐⇒

v1=0 v2 =0. So, condition (3.1) is satisfied. Similarly, (3.1) holds for everyx¯ ∈

(x₁,4) : x₁ ≥ 2 . In addition, for everyv = (v1,0) ∈ (0⁺K)\{0}, one hasb₁^Tv = b₂^Tv = v1 > 0. Thus, condition (3.2) is satisfied. Therefore,

E^Ge=E=E^w=

(x₁,0) : x₁≥2} ∪ {(x₁,4) : x₁≥2 .

4.2 LFVOPs having more than two connected components in the solution sets

To proceed furthermore, we need the following specific result.

Proposition 4.2 For a LFVOP in the form(VP), if the denominators of the functions f_iare all the same, then one has E^Ge=E.

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Proof Suppose that the denominators of f_i coincide. To show that E = E^Ge, we fix any

¯

x ∈ E. By the necessary and sufficient optimality condition recalled in (2.1), there is a multiplierξ =(ξ1, . . . , ξm)∈intR^m₊such that

^m

i=1

ξi

b_i^Tx¯+βi

a_i−

a_i^Tx¯+αi)b_i ,y− ¯x

≥0, ∀y∈K. (4.1)

For eachi ∈I, one has ∇f_i(x¯),y− ¯x

= 1 p(x¯)

b_i^Tx¯+βi

a_i−

a_i^Tx¯+αi)b_i,y− ¯x , wherep(x¯):=

b^T₁x¯+β1

₂

= · · · =

b_m^Tx¯+βm

₂

(see the first equality in [23, p. 147]).

Hence, we can rewrite (4.1) equivalently as ^m

i=1

ξip(x¯)∇fi(x¯),y− ¯x

≥0, ∀y∈K.

Puttingξ¯i =ξip(x¯)and noting thatξ¯i >0 fori ∈I, we get ^m

i=1

ξ¯i∇fi(x¯),y− ¯x

≥0, ∀y∈K. (4.2)

Clearly, (4.2) shows thatx¯is satisfies the first-order necessary optimality condition for the following scalar optimization problem

(P0) Minimize ϕ(x):=

m i=1

ξ¯ifi(x) subject to x∈K.

It is easy to verify that, in this example,ϕ(x)is a linear fractional function. Since∇ϕ(x)¯ = m

i=1

ξ¯i∇fi(x¯), combining (4.2) with (2.3) yieldsϕ(y)≥ϕ(x¯)for ally∈K. This means that

¯

xis a global solution of (P0). Then, by [13, Theorem 1] we can assert thatx¯∈E^Ge.

The proof is complete.

The efficient solution set of the problem considered in Example4.1has two connected components. We now analyze a LFVOP whose efficient solution set has three connected components.

Example 4.3 (See [17, p. 483]) Consider problem(VP)wheren=m=3, K =

x∈R³ :x1+x2−2x3≤1, x1−2x2+x3≤1,

−2x₁+x₂+x₃≤1, x₁+x₂+x₃≥1 , and

fi(x)= −xi+1 2 x1+x2+x3−3

4

(i=1,2,3).

Here we have

a₁=(−1,0,0), a₂=(0,−1,0), a₃=(0,0,−1), α1=α2=α3= 1 2,

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and

b₁ =b₂=b₃=(1,1,1), β1=β2=β3= −3 4. Setting

C =

⎛

⎜⎜

⎝

1 1 −2 1 −2 1

−2 1 1

−1 −1 −1

⎞

⎟⎟

⎠, d=

⎛

⎜⎜

⎝ 1 1 1

−1

⎞

⎟⎟

⎠,

we see thatK = {x ∈R³ : C x≤d}.As it has been shown in [17], E=E^w= {(x₁,x₂,x₃) : x₁ ≥1, x₃=x₂=x₁−1}

∪{(x₁,x₂,x₃) : x₂≥1, x₃=x₁=x₂−1}

∪{(x1,x2,x3) : x3≥1, x2 =x1=x3−1}.

(4.3)

Settingp(x)=

x1+x2+x3−3 4 2

, one has

∇f₁(x)= 1 p(x)

−x₂−x₃+1 4,x₁− 1

2,x₁−1 2

,

∇f₂(x)= 1 p(x)

x₂−1

2,−x₁−x₃+1 4,x₂−1

2

,

∇f₃(x)= 1 p(x)

x₃−1

2,x₃−1

2,−x₁−x₂+1 4

.

By Lemma2.9, we get 0⁺K = {v=(τ, τ, τ)∈R³ : τ≥0}. Asb^T_i v=0 for anyi ∈Iand v∈0⁺K\{0}, the second regularity condition (3.2) is satisfied for everyx¯ ∈ E. To check the first and the third regularity conditions in (3.1) and (3.10), fix anyx¯ =(x¯₁,x¯₂,x¯₃)∈ E withx¯₁≥1 andx¯₂= ¯x₃ = ¯x₁−1. Note that

∇f1(x)¯ = 1 p(x¯)

−2x¯1+9 4,x¯1−1

2,x¯1−1 2

,

∇f2(x)¯ = 1 p(x¯)

¯ x1−3

2,−2x¯1+5 4,x¯1−3

2

,

∇f3(x)¯ = 1 p(x¯)

¯ x₁−3

2,x¯₁−3

2,−2x¯₁+5 4

. Hence, we have

∇f1(x¯), v = 5τ

4p(x¯) >0, ∇f2(x¯), v = − 7τ

4p(x)¯ <0,∇f3(x), v = −¯ 7τ 4p(x¯) <0 for anyv=(τ, τ, τ)withτ >0. Therefore, both conditions (3.1) and (3.10) are satisfied.

Thus, by Theorem3.2we can assert thatx¯∈E^Ge. The fact that the last inclusion holds for anyx¯∈Efollows from the description ofEin (4.3) and the symmetry of our problem(VP) w.r.t. the variablesx1,x2,x3. This means thatE=E^Ge.

Remark 4.4 In Example 4.3, since the denominators of fi are all the same fori ∈ I, by Proposition4.2we haveE^Ge=E =E^w. This result fully agrees with the one obtained by using Theorem3.2.

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Now, let us examine a LFVOP where the number of criteria can be any integerm≥2.

Example 4.5 (See [17, pp. 479–480]) We consider problem(VP)wheren=m,m≥2,

K =

x∈R^m : x1≥0, x2≥0, . . . , xm≥0, m k=1

xk≥1 ,

and

fi(x)= −xi+1 _m 2

k=1x_k−³₄ (i=1, . . . ,m).

Here we have

a₁=(−1,0,0, . . . ,0), a₂=(0,−1,0, . . . ,0), . . . , am=(0,0,0, . . . ,−1), α1=α2= · · · =αm= 1

2, and

b₁=b₂= · · · =b_m=(1,1,1, . . . ,1), β1=β2= · · · =βm= −3 4. Setting

C=

⎛

⎜⎜

⎜⎝

−1 0 0 . . . 0

0 −1 0 . . . 0

0 0 −1 . . . 0

... ... ... ...

0 0 0 . . . −1

−1 −1 −1 . . . −1

⎞

⎟⎟

⎟⎠

, d=

⎛

⎜⎜

⎜⎝ 0 0 0...

0

−1

⎞

⎟⎟

⎟⎠ ,

we see thatK = {x ∈Rⁿ : C x ≤d}.According to [17, p. 483], one has E=E^w= {(x₁,0, . . . ,0) : x₁≥1}

∪{(0,x₂, . . . ,0) : x₂≥1} . . . .

∪{(0, . . . ,0,xm) : xm≥1}.

(4.4)

Sinceb_i = (1,1,1, . . . ,1)^T fori = 1, . . . ,m and 0⁺K = R^m₊, ifb_i^Tv = 0 for some v ∈0⁺K, thenv =0. This implies that the second regularity in (3.2) is satisfied. Setting q(x)=

_m

k=1

xk−3 4

2

, one has

∇fi(x)= 1 q(x)

x_i−1

2, . . . ,−

k=i

x_k+1

4, . . . ,x_i−1 2